Problem 13 - Algebra and Trigonometry

Question

Find all solutions of each equation. $$ \tan x=1 $$

Step-by-Step Solution

Answer

The solutions to the given trigonometric equation are \(x = \frac{\pi}{4} + n\pi\) and \(x = \frac{5\pi}{4} + n\pi\) where \(n\) is any integer

Step 1: Identify the initial solutions

First, recall the unit circle. The \(\tan\) function is equal to 1 at \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\) in the interval \([0, 2\pi]\). So these two become our initial solutions.

Step 2: Identify the period of the function

Next, know that the period of the \(\tan\) function is \(\pi\) or 180 degrees.

Step 3: Find all solutions

Since tan is periodic, if \(x = a\) is a solution then \(x = a + n\pi\) is also a solution for any integer \(n\). So, the solutions are given by \(x = \frac{\pi}{4} + n\pi\) and \(x = \frac{5\pi}{4} + n\pi\), for all integer values of \(n\).

Key Concepts

Tangent Function

The tangent function, often written as \(\tan(x)\), is a fundamental trigonometric function that plays a crucial role in various mathematical applications.
It is formed by dividing the sine function by the cosine function: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This division leads to the unique characteristic that \(\tan(x)\) can be undefined, particularly when \(\cos(x) = 0\). These points of undefined values occur at odd multiples of \(\frac{\pi}{2}\).

• Tangent is often connected to the slope of a line in a coordinate plane, providing a measure of steepness.
• Unlike cosine and sine, the tangent function has a range of all real numbers, which means it can take on any real value.
• The graph of \(\tan(x)\) starts repeating its pattern after a particular interval.

These features make understand the tangent function critical when solving trigonometric equations.
To solve \(\tan x = 1\), knowing that \(\tan(x) = 1\) corresponds to a slope of 1, essential in finding the solutions on the unit circle where the angle has a tangent of 1.

Unit Circle

The unit circle is an indispensable tool in trigonometry that helps visualize and solve trigonometric equations. It is a circle with a radius of 1 centered at the origin \((0,0)\) of a coordinate plane.
The unit circle allows us to easily find sine, cosine, and tangent values for angles.

• Within the unit circle, every point \((x, y)\) on the circle translates to \((\cos(\theta), \sin(\theta))\) where \(\theta\) is the angle formed with the positive x-axis.
• For tangent, it is the ratio of the \(y\)-coordinate to the \(x\)-coordinate: \(\tan(\theta) = \frac{y}{x}\).
• Key angles such as \(\frac{\pi}{4}\) or 45 degrees, where \(\tan(x) = 1\), occur on the unit circle.

The main idea is that by identifying key angles on the unit circle, we can solve equations involving tangent and other trigonometric functions.
For \(\tan x = 1\) specifically, these angles would be at \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\). This geometric perspective helps in understanding periodicity as it visually reinforces how these points repeat around the circle.

Periodicity of Functions

Periodicity is a core concept of trigonometric functions, meaning they repeat their values at regular intervals.
The tangent function is especially notable for its periodicity.

• \(\tan(x)\) has a period of \(\pi\), which is significantly different from sine and cosine, both of which have a period of \(2\pi\).
• This property translates to every change in \(\pi\) radians or 180 degrees resulting in the function returning to the same value it had before.
• Because of periodicity, once an initial solution is found, like \(\frac{\pi}{4}\) for \(\tan x = 1\), solutions can be generated endlessly by adding \(n\pi\), where \(n\) is an integer (\(n\pi\) accounts for one complete cycle).

In solving \(\tan x = 1\), periodicity is why additional solutions like \(\frac{5\pi}{4} + n\pi\) can be found, emphasizing the function's predictable repeating behavior.
Understanding periodicity simplifies addressing trigonometric equations by anticipating where and how often solutions will emerge.